A Study On The Infectious Disease Model Of A Discontinuous Treatment Strategy

The outbreak of unexpected contagious disease SARS makes people pay more attention to the the importance of the capacity of the health care system for the disease. Some scholars consider introducing the piecewise therapeutic function of treatment rates in the traditional function of the model. It is assumed that treatment rate of the disease is proportional to the number of infected persons below the capacity and is a constant when the number of infected persons is greater than the capacity.Firstly, mathematical modeling of infectious disease dynamics system, the infection rate function will vary with the size of the di?erent population size, general incidence coincide with the actual. Due to the di?erent ?ows of residents and communities of medical resources, therefore,treatment of the disease will be limited to the maximum capacity of the health care system. Considering these circumstances, establishing the SIS epidemic model with the general infection rate function and treatment. It is found that,there exists a diseasefree equilibrium which is locally asymptotically stable when the basic reproduction number is less than 1. When the basic reproductive number is greater than 1,the system has a unique positive equilibrium point, which is locally asymptotically stable if the infected number below the capacity of the medical system; if the infected number exceeds the maximum capacity of the medical system, the system may exist two positive equilibria or no positive equilibrium when the basic reproduction number is less than 1. If there are two positive equilibria, then endemic equilibrium with smaller infected number is a saddle point, and the other endemic equilibrium is a stable node or focus. Furthermore, it is found that system undergoes a backward bifurcation when the treatment capacity of the health system is weaker.Secondly, we considered the SEIS epidemic model with saturation incidence and piecewise treatment function. It is found that,if the number of infected persons below the capacity of the medical system, the disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than 1, a unique epidemic equilibrium of the system is globally asymptotically stable when the basic reproduction number is greater than 1; if the number of infected persons is greater than the capacity of the medical system, the system may exist two endemic equilibria. Finally, in order to coincide with the actual problems more accurately and reduce the cost of controlling the disease, introducing the discontinuous treatment strategies in the traditional model of infectious diseases under consideration,establishing an SIR epidemic model with discontinuous treatment strategies and saturated incidence. According to the right of discontinuous solutions of di?erential equations of theoretical knowledge to get the existence of Filippov solution in the model. It is found that when the basic reproductive number is not greater than 1, the disease-free equilibrium is globally asymptotically stable; when the basic reproductive number is greater than 1, the positive equilibrium is globally asymptotically stable under some reasonable assumptions.In addition, it is found that the model converge to the disease-free equilibrium point within a limited time under some reasonable assumptions. Thus you can change some parameters to cut down the treatment time of the disease, it is very important for the medical sta?s to develop the appropriate health strategies which eliminate the disease in ?nite time.